# Properties

 Label 2178.2.a.g Level $2178$ Weight $2$ Character orbit 2178.a Self dual yes Analytic conductor $17.391$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2178 = 2 \cdot 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2178.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$17.3914175602$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 66) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - 2q^{5} + 4q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} - 2q^{5} + 4q^{7} + q^{8} - 2q^{10} + 6q^{13} + 4q^{14} + q^{16} + 2q^{17} - 4q^{19} - 2q^{20} - 4q^{23} - q^{25} + 6q^{26} + 4q^{28} + 6q^{29} + q^{32} + 2q^{34} - 8q^{35} + 6q^{37} - 4q^{38} - 2q^{40} - 6q^{41} - 4q^{43} - 4q^{46} + 12q^{47} + 9q^{49} - q^{50} + 6q^{52} - 2q^{53} + 4q^{56} + 6q^{58} - 12q^{59} + 14q^{61} + q^{64} - 12q^{65} + 4q^{67} + 2q^{68} - 8q^{70} + 12q^{71} + 6q^{73} + 6q^{74} - 4q^{76} + 4q^{79} - 2q^{80} - 6q^{82} + 4q^{83} - 4q^{85} - 4q^{86} - 10q^{89} + 24q^{91} - 4q^{92} + 12q^{94} + 8q^{95} - 14q^{97} + 9q^{98} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 0 1.00000 −2.00000 0 4.00000 1.00000 0 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2178.2.a.g 1
3.b odd 2 1 726.2.a.c 1
11.b odd 2 1 198.2.a.a 1
12.b even 2 1 5808.2.a.bc 1
33.d even 2 1 66.2.a.b 1
33.f even 10 4 726.2.e.g 4
33.h odd 10 4 726.2.e.o 4
44.c even 2 1 1584.2.a.f 1
55.d odd 2 1 4950.2.a.bu 1
55.e even 4 2 4950.2.c.p 2
77.b even 2 1 9702.2.a.x 1
88.b odd 2 1 6336.2.a.bw 1
88.g even 2 1 6336.2.a.cj 1
99.g even 6 2 1782.2.e.e 2
99.h odd 6 2 1782.2.e.v 2
132.d odd 2 1 528.2.a.j 1
165.d even 2 1 1650.2.a.k 1
165.l odd 4 2 1650.2.c.e 2
231.h odd 2 1 3234.2.a.t 1
264.m even 2 1 2112.2.a.r 1
264.p odd 2 1 2112.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.a.b 1 33.d even 2 1
198.2.a.a 1 11.b odd 2 1
528.2.a.j 1 132.d odd 2 1
726.2.a.c 1 3.b odd 2 1
726.2.e.g 4 33.f even 10 4
726.2.e.o 4 33.h odd 10 4
1584.2.a.f 1 44.c even 2 1
1650.2.a.k 1 165.d even 2 1
1650.2.c.e 2 165.l odd 4 2
1782.2.e.e 2 99.g even 6 2
1782.2.e.v 2 99.h odd 6 2
2112.2.a.e 1 264.p odd 2 1
2112.2.a.r 1 264.m even 2 1
2178.2.a.g 1 1.a even 1 1 trivial
3234.2.a.t 1 231.h odd 2 1
4950.2.a.bu 1 55.d odd 2 1
4950.2.c.p 2 55.e even 4 2
5808.2.a.bc 1 12.b even 2 1
6336.2.a.bw 1 88.b odd 2 1
6336.2.a.cj 1 88.g even 2 1
9702.2.a.x 1 77.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2178))$$:

 $$T_{5} + 2$$ $$T_{7} - 4$$ $$T_{13} - 6$$ $$T_{17} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$2 + T$$
$7$ $$-4 + T$$
$11$ $$T$$
$13$ $$-6 + T$$
$17$ $$-2 + T$$
$19$ $$4 + T$$
$23$ $$4 + T$$
$29$ $$-6 + T$$
$31$ $$T$$
$37$ $$-6 + T$$
$41$ $$6 + T$$
$43$ $$4 + T$$
$47$ $$-12 + T$$
$53$ $$2 + T$$
$59$ $$12 + T$$
$61$ $$-14 + T$$
$67$ $$-4 + T$$
$71$ $$-12 + T$$
$73$ $$-6 + T$$
$79$ $$-4 + T$$
$83$ $$-4 + T$$
$89$ $$10 + T$$
$97$ $$14 + T$$